The northern route through southern Russia to Hungary is pretty obvious: it’s a continuous band of steppes that would appeal to pastoralists. But south of the Caspian, I don’t see the overwhelming appeal of following the mountains to Anatolia as opposed to following the Fertile Crescent into Egypt (as the Mongols attempted), or following the Persian Gulf into southern Arabia.
Well, they didn't, at least not when they migrated. The Seljuq (Or Seljuk) Empire was a Turkish state that dominated what we now call the Middle East, seen as its greatest extent here. Although it can be more accurately described as Turko-Persian due to its reliance upon the existing Persian system (Much like the Yuan Dynasty in China), they came to power in Persia after a number of great upheavels, originally as allied soldiers and eventually establishing their own power base.
They only lasted as a unified state until 1092, whereupon the very familiar succession politics that plague many states ended up fracturing the Empire. While Turks as a people mostly disappeared from many of these areas, many Turks settled in the North-west of the Empire. This section of the Empire, during the succession crisis I mentioned earlier, became known as the Sultanate of Rum, meaning Roman, and was located in the parts of Anatolia that the Seljuks had conquered from the Byzantines.
This settlement was partially to defend the frontiers against Georgian incursions. This leads into Turkish history where Rum eventually shrivels due to the Mongol invasion and fractures, leaving independent beyliks to eventually found the Ottoman Empire, but suffice it to say, the reason for the denseness of the Turkish population in Anatolia is due to political decisions taken by the Seljuks. Of course there could be other reasons as well, and there likely are, but that is outside of my area and this is very likely one of the main reasons why the Turkic tribes eventually dominated Anatolia.
I'm not an expert (and I'm on a mobile), but I believe that the migration was more by chance than by planning.
By the time of the Turkish migration in the 10th and 11th centuries into the Middle East, the Sunni Abbasid Caliphate (750-1258), which had been the preeminent force through much of the Middle East and North Africa was struggling against internal strife and a civil war of sorts against the rising strength of the Shiite Fatimid Caliphate ( 909-1171) based in Cairo. In this condition of decline, the Abbasid leadership were unable to withstand the incursions of the superior Turkish forces. The Turks, much like the Mongols who would follow, were adept with tactics and masters of mounted archery. Unlike the Mongols, the Turks adapted to the culture around them and converted to Islam integrating (somewhat) into Muslim society and eventually becoming the power of the Abbasid military. This breathed a whole new life into the Caliphate, though most Muslims despised the Turks and held their version of Islam in contempt. To keep the Turks occupied whilst separating them from the population, the Abbasid leadership sent the Turkish raiders to the Anatolian border of the Byzantine Empire, a powerful realm overflowing with wealth....
...Except the Empire was broke and in confusion. In 1025, the Emperor ( or Basileus) Basil II died without an heir (besides his aged brother and his nieces who were beyond childbearing years) effectively ending the Macedonian dynasty and a renaissance that spanned two centuries. What followed was a succession of weak emperors who failed to maintain Basil's carefully arranged border defenses, squandered the Empire's wealth, and divided the realm. As the Turks encroached on Byzantine territory, they encountered little to no resistance. Finally, the Byzantines rallied under Basileus Romanos IV Diogenes and moved to confront the raiders, engaging at Manzikert in 1071. The ensuing rout that the Byzantines suffered resulted in the obliteration of their army, the capture of the Basileus Romanos IV, and the permanent loss of Asia Minor. The other effect was the rise of the Komnenoi.
Fast forward through the call for aid by Basileus Alexios Komnenos which helped precipitate the crusades, the First Crusade, and stop in the reign of Manuel I in 1146. The economy of the Byzantine Empire was recovering and Manuel was leading a successful campaign against the Turks based in Konye and styled the Sultanate of Rum (Rome). However, upon learning that the Second Crusade had commenced and that European forces would soon be at Constantinople, Manuel abandoned the campaign and returned to the capital to receive the Crusaders and weave through the complicated politics of Eastern and Western Europe. In 1176, Manuel would again set his sights on the Turks. The clumsy campaign would culminate at Myriokephalon in a disaster similar to Manzikert. This defeat would be the last great attempt to expel the Turks from Asia Minor.
Fast forwarding again to 1204, Constantinople is THOROUGHLY sacked by the Crusaders. There are many different reasons and conspiracy theories as to why this took place which are not pertinent to your question, but the point is that after 1204, Constantinople is in a constant state of decline and never again able effectively raise resistance against the Turkish invaders. This decline would greatly aid Turkish hegemony in the region and the eventual conquest of Constantinople by Mahomet the Conqueror in 1453.
It should also be noted that during most of this period, the Turks and the Byzantine citizens on the Anatolian border were active in trading and that a certain amount of cohabitation existed.
tl; dr a combination of Abbasid reassignment, Byzantine decay, Latin interference, Turkish military strength, and good ol' fashioned luck led the Turks to Anatolia.
Sources: God's War by Christopher Tyerman
"The Era of the Crusades" audiobook lecture by Professor Kenneth W. Harl courtesy of Audible
